If the function f : I → R is differentiable on the interval I ⊆ R , then for each x,a ∈ I, according to the mean value theorem, there exists a number c(x) belonging to the open interval determined by x and a , and there exists a real number θ (x) ∈]0,1[ such that f (x) − f (a) = (x − a) f (1) (c(x)) and f (x) − f (a) = (x − a) f (1) (a + (x − a)θ (x)). In this paper we shall study the differentiability of the functions c and θ in a neighbourhood of a. Mathematics subject classification (2000): 26A24.

Abstract. In this paper, we generalize identity (3), from where we obtain a rafinement of inequalities (1) and (2). IntroductionThe inequality from Theorem 1 is called in the literature Bergström's inequality (see [2]with equality if and only ifThe generalization of Bergström's inequality is contained in the following theorem (see [5]). THEOREM 2. If x k ∈ R and a k > 0 , k ∈ {1, 2,...,n} , thenBy particularizations in Theorem 2, in paper [5] the refinements are obtained of Cauchy-Schwarz's inequality.For the complex numbers, it is well-known the identity:Mathematics subject classification (2000): 26D15.

If the functions f , g : I → R are differentiable on the interval I ⊆ R , then for each x, a ∈ I there exists a real number θ ∈]0, 1[ such that (f (x) − f (a)) g (1) (a + θ(x − a)) = (g (x) − g (a)) f (1) (a + θ(x − a)). In this paper we study the behaviour of the number θ ∈]0, 1[, when x approaches a .

No abstract

Abstract. In [13], we study a class of linear and positive operators defined by finite sum. In this paper we demonstrate general properties for a class of linear positive operators defined by infinite sum. By particularization, we obtain statements, the convergence and the evaluation for the rate of convergence in therm of the first modulus of smoothness for the Mirakjan-Favard-Szasz operators, Baskakov operators and Mayer-Konig and Zeller operators. We don't study the convergence of these operators with the well known theorem of Bohman-Korowkin. IntroductionIn this section, we recall some notions and results which we will use in this axticle.Let